1. Field of the Invention
This invention relates generally to methods of characterizing and modeling erbium doped fiber amplifiers (EDFA). More specifically, this invention relates to methods for modeling EDFA having erbium doped fiber (EDF) sections of arbitrary length using a general purpose digital computer to determine at least the average inversion level and output power of the EDFA.
2. Description of the Related Art
Fiber optic communication methods and hardware have become ubiquitous in our modem age for transporting large amounts of data from one location to another. Fiber optics utilize a light wave as a carrier signal and a modulated light signal to transport the particular information or data to be sent and received. Fiber optic communication systems typically utilize a coherent light source, such as a laser, to provide the carrier and modulation frequencies.
Fiberoptic communication also allows for efficient multiplexing of signals. Multiplexing refers to techniques that are used to pack more information down the single fiber or xe2x80x9cpipexe2x80x9d by simultaneously transmitting several signals over the same pipe. Each signal is uniquely tagged so that the receiver can recognize the different signals. One important and frequently implemented multiplexing technique is called xe2x80x9cwavelength-division multiplexingxe2x80x9d (WDM), which accomplishes this result by delivering each such signal on a slightly different laser frequency that is detected by the receiver. Each slightly different frequency signal is called a xe2x80x9cchannel.xe2x80x9d
The challenge in current fiber optic communication systems is to efficiently implement multi-channel, high data rate and/or long distance systems. In long distance systems (sometimes referred to as xe2x80x9clong haul systemsxe2x80x9d) and high data rate systems, it is often a challenge to maintain the modulated signals"" integrity because the silica or glass that makes up the fiber optic pipe exhibits high loss and high dispersion characteristics at the wavelength of the light source. The optical properties of silica pipes require that long haul and/or high data rate systems operate at between 1.33 and 1.55 xcexcm. Moreover, the critical challenge for long haul fiber optic communications over distances of more than 100 km is how to efficiently and effectively amplify the optical signals. Typically, repeaters placed at regular intervals along the fiber link have been used to amplify the signals. Repeaters are electrooptic devices that first convert the attenuated light into an electrical signal that is then amplified and regenerated. A diode laser in the repeater transforms the electrical signal back into light for further transmission down the fiber. The problem with repeaters is that their complex circuitry fixes the electronic speed of each device, making upgrades difficult and costly. Repeaters also tend to complicate WDM systems and increase their cost because the channels must be amplified one at a time.
In the middle 1980""s, researchers at the University of Southampton (Southampton, England) developed an all-optical way to amplify the 1.55 xcexcm light in long haul fiberoptic systems. Recognizing that the rare earth element erbium exhibits good metastable states at the appropriate frequencies, they doped a three meter long silica fiber core with erbium and optically pumped it at 650 nm. In this fashion they were able to generate 125 dB of gain for a 1.53 xcexcm signal. This was the first EDFA which today has become the optical amplifier of choice especially in all-optical, terrestrial and transoceanic fiber links.
Commercial EDFAs are typically end-pumped by a semiconductor laser at either 980 or 1480 nm. The pump radiation is introduced to the core by a dichroic coupler (a beam splitter), leaving the signal and pump waves to travel through the core together. Because the narrow core keeps the pump radiation concentrated in a small volume, it takes only a few milliwatts of pump power to generate good gain at the signal wavelength. To avoid unwanted resonance absorption of the signal by unexcited erbium atoms, the entire length of the doped fiber through which the signal travels must be pumped.
Various methods have been developed in the past in attempts to analyze the gain, output power, amplified spontaneous emission, noise figures and other parameters of EDF in order to characterize EDFAs and produce better devices. Recently, an xe2x80x9caverage inversion modelxe2x80x9d has been proposed which investigates the average inversion level of erbium atoms over a length of EDF to characterize the EDF. See Y. Sun et al., xe2x80x9cAverage Inversion Level, Modeling and Physics of Erbium-Doped Fiber Amplifiersxe2x80x9d, IEEE J. STQE, Vol. 3, No. 4, pp. 991-1007 (1997). Many other models have been proposed for investigating EDF, all of which require rigorous solutions to multi-variable partial differential equations which do not result in exact expressions, and which are computationally complex so that the models require advanced computers and a lot of processor time to solve. Moreover, as the need has evolved for ever increased bandwidth, the number of channels sent through the pipe has grown. This need has given rise to a multiplexing scheme called xe2x80x9cdense wavelength division multiplexingxe2x80x9d (DWDM) which has not been well defined by the current models.
There thus exist long-felt but unsolved needs in the art for models of EDF and EDFA that can accurately characterize these devices mathematically. Such models should simply explain the dynamics of the inversion population of the erbium atoms at all points in the pipe, and produce meaningful output power expressions at any point along the pipe. Additionally, the models should be computationally simple so as to not dominate computer processor time when they are being run on the computer.
The aforementioned long-felt needs are solved, and problems met, by the methods provided in accordance with the present invention. The invention comprises a model of an EDF having a length and a fractional population density of erbium ions in an excited state. Since the EDF supports N channels, the power propagation along the EDF is characterized by N+1 differential equations as a function of the direction of propagation, z, of the channels along the length l of the EDF and the time t. By applying an average inversion model to a spatially averaged inversion level of the erbium ions in the fiber, the N+1 partial differential equations are reduced to a single ordinary differential equation. This allows an analytical solution at the boundary and initial conditions of the fiber so that an expression for the power of the signal propagating along the fiber can be obtained. With this expression, the single equation can be solved analytically for the inversion level at any point along the EDF. The inventive model provides solutions to the inversion level and therefore many other useful parameters can be obtained for both the steady state and time-dependent cases.
These and other features of the present invention will become apparent from the following detailed description considered in conjunction with the accompanying drawings. It is to be understood, however, that the drawings are designed solely for purposes of illustration and not as a definition of the invention, for which reference should be made to the appended claims.